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Distribution of the scan statistic for a sequence of bistate trials

Part of: Distribution theory - Probability Markov processes

Published online by Cambridge University Press:  14 July 2016

James C. Fu
James C. Fu*
Affiliation:
University of Manitoba
*
∗ Postal address: Department of Statistics, University of Manitoba, Winnipeg, Manitoba, Canada R3T 2N2. Email address: fu@ccu.umanitoba.ca
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Abstract

Let be the scan statistic of window size r for a sequence of n bistate trials . The scan statistic Sn(r) has been successfully used in various fields of applied probability and statistics, and its distribution has been studied extensively in the literature. Currently, all existing formulae for the distribution of Sn(r) are rather complex, and they can only be numerically implemented when is a sequence of Bernoulli trials, the window size r is less than 20 and the length of the sequence n is not too large. Hence, these formulae have been limiting the practical applications of the scan statistic. In this article, we derive a simple and effective formula for the distribution of Sn(r) via the finite Markov chain embedding technique to overcome some of the limitations of the existing complex formulae. This new formula can be applied when is either a sequence of Bernoulli trials or a sequence of Markov dependent bistate trials. Selected numerical examples are given to illustrate our results.

Keywords

Compound patternwaiting time distributionfinite Markov chain embeddingtransition probability matrix

MSC classification

Primary: 60E05: Distributions
Secondary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Type
Research Papers
Information
Journal of Applied Probability , Volume 38 , Issue 4 , December 2001 , pp. 908 - 916
Copyright
Copyright © by the Applied Probability Trust 2001 

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Footnotes

This work was supported in part by Grant A-9216 of the Natural Science and Engineering Research Council of Canada.

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